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Dr. Shurjeel Wyne delivered this lecture at COMSATS Institute of Information Technology, Attock for Digital Communication Systems course. In this he discussed: Encoder, Convolutional, Codes, Channel, Encoder, Shift, Register, Effective, Rate, Vector, Representation
Typology: Slides
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Today, we are going to talk about:
Another class of linear codes, known as Convolutional codes.
We study the structure of the encoder.
We study different ways for representing the encoder.
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Convolutional Codes
In convolutional coding, the channel encoder maps a continuous sequence of information bits into a continuous sequence of encoded output bits Convolutional coding differs from block coding in that information bits are not grouped into distinct blocks for encoding, rather the entire data stream can be encoded into a single codeword Convolutional codes can achieve a larger coding gain as compared to block codes with the same complexity The convolutional encoder requires memory elements, a code is generated by passing the information bit sequence through a finite state shift register
modulo-2 adders
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Convolutional codes-cont’d
is the coding rate, which determines the number of data bits per coded bit. In practice, usually k=1 is chosen and we will assume k=1 in this course (unless stated otherwise).
K is the constraint length of the encoder, where the encoder has ( K-1)k memory elements. The constraint length represents the number of k -bit shifts over which a single information bit can influence the encoder output.
For the k=1 case that we will assume in this course, this means that the encoder has (K-1) memory elements and each information bit will influence encoder output over K single- bit shifts.
( n , k , K ) ( k / n , K )
k / n
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Block diagram of the DCS
Information source
Rate 1/n Conv. encoder Modulator
Information sink
Rate 1/n Conv. decoder Demodulator m ˆ^ ( m ˆ 1 , m ˆ 2 ,..., m ˆ i ,...)
(^1) Input (^2) sequence Channel m ( m , m ,..., mi ,...)
Branch word( coded bits)
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Codeword sequence
( 1 , 2 , 3 ,..., ,...)
n
i i ji ni
i
U u ,...,u ,...,u
U U U U
U ^ G(m)
outputsperBranch wor d
1 forDemodulato Branch worroutputsd
receivedsequence
( 1 , 2 , 3 ,..., ,...)
n
i ji ni i
i
Zi z,...,z ,...,z
Z Z Z Z
Z
i = time index
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Rate ½ Convolutional encoder
Input data bits Output coded bits m
u 1
u 2
First coded bit
Second coded bit
u 1 , u 2
(Branch word)
3 shift-registers where the first one takes the incoming data bit and the rest, form the memory of the encoder.
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Rate ½ Convolutional encoder –
Cont’d
t 1 1 0 0
u 1
u 2
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u 1 u (^2) t 2 0 1 0
u 1
u 2
10
u 1 u 2
t 3 1 0 1
u 1
u 2
00
u 1 u (^2) t 4 0 1 0
u 1
u 2
10
u 1 u 2
m ( 101 ) Time (^) Output Time Output
Message sequence:
(Branch word) (Branch word)
msg bits are input at times t1, t2, t
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Encoder representation
Vector representation: We define n binary K-vectors (one vector for each modulo- 2 adder). The i :th element in each vector, is “1” if the i :th stage in the shift register is connected to the corresponding modulo-2 adder, and “0” otherwise. Example:
m
u 1
u 2 ( 101 ) u^1 u^2
( 111 ) 2
1
g
g
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Encoder representation – cont’d
Polynomial representation: We define n generator polynomials, one for each modulo- adder. Each polynomial is of degree K-1 or less and its coefficients are either 0 or 1 depending on whether or not the shift register is connected to the corresponding modulo-2 adder. Example:
The output sequence is found as follows:
( 2 ) 2 2 2
( 2 ) 1
( 2 ) 2 0
( 1 ) 2 2 2
( 1 ) 1
( 1 ) 1 0 ( ).. 1
X g g X g X X
X g g X g X X X
g
g
U ( X ) m ( X ) g 1 ( X )interlacedwith m ( X ) g 2 ( X )
( 101 )
( 111 ) 2
1
g
g
Equivalent Vector representation
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Encoder representation –cont’d
Details of Polynomial representation:
2 3 4
2 3 4 2
2 3 4 1
2 2 4 2
2 2 3 4 1
m g
m g
m g
m g
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State diagram
A convolutional encoder belongs to a class of devices known as finite-state machines A finite-state machine only encounters a finite number of states. State of a machine: the smallest amount of information that, together with a current input to the machine, can predict the output of the machine. In a Convolutional encoder, the state is represented by the contents of the memory. Hence, there are 2(K-1)^ states.
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Trellis Diagram
Example of a section of trellis for the rate ½ code
t (^) i ti 1 Time
State S 0 00
0/
1/
0/
0/ 0/
1/
1/
1/
